Some Properties of the Exit Measure for Super-brownian Motion

We consider the exit measure of super-Brownian motion with a stable branching mechanism of a smooth domain D of R d. We derive lower bounds for the hitting probability of small balls for the exit measure and upper bounds in the critical dimension. This completes the results given by Sheu 20] and generalizes the results of Abraham and Le Gall 2]. We give also the Hausdorr dimension of the exit measure and show it is totally disconnected in high dimension. Eventually we prove the exit measure is singular with respect to the surface measure on @D in the critical dimension. Our main tool is the subordinated Brownian snake introduced First we introduce some notation. We denote by (M f ; M f) the space of all nite nonneg-ative measures on R d , endowed with the topology of weak convergence. We denote by B(R d) the set of all measurable functions deened on R d taking values in R. With a slight abuse of notation, we also denote by B(R d) the Borel-eld on R d. For every measure 2 M f , and every nonnegative function f 2 B(R d), we shall use both notations R f(y)(dy) = (; f). We also write (A) = (; 1 A) for A 2 B(R d). We write supp for the closed support of a measure 2 M f. If A 2 B(R d), then A denotes the closure of A. Let d 2. Let 2 (1; 2]. Let be a Brownian motion in R d started at x under P x .